Certification Problem

Input (TPDB SRS_Standard/ICFP_2010/5130)

The rewrite relation of the following TRS is considered.

0(0(1(2(x1)))) 3(4(5(4(4(5(3(0(0(3(x1)))))))))) (1)
3(2(2(2(x1)))) 0(3(2(4(4(4(1(3(5(3(x1)))))))))) (2)
0(2(4(5(2(x1))))) 0(3(4(0(0(0(0(3(1(3(x1)))))))))) (3)
1(4(0(5(1(x1))))) 5(4(4(4(5(4(3(3(3(0(x1)))))))))) (4)
5(4(1(2(0(x1))))) 0(0(4(4(3(0(4(1(3(0(x1)))))))))) (5)
0(3(2(5(3(5(x1)))))) 3(1(3(4(4(3(4(3(3(5(x1)))))))))) (6)
0(5(1(1(5(0(x1)))))) 0(0(0(3(1(3(3(1(3(0(x1)))))))))) (7)
1(2(5(2(2(3(x1)))))) 0(4(2(4(5(4(3(4(3(3(x1)))))))))) (8)
2(0(0(1(2(3(x1)))))) 1(3(3(3(1(3(4(4(4(3(x1)))))))))) (9)
2(0(1(3(5(1(x1)))))) 0(0(3(4(5(4(5(3(0(1(x1)))))))))) (10)
2(5(5(0(0(1(x1)))))) 4(1(0(3(4(3(1(5(3(0(x1)))))))))) (11)
3(0(4(0(3(1(x1)))))) 4(4(4(5(3(0(1(0(3(1(x1)))))))))) (12)
3(2(1(5(5(0(x1)))))) 4(1(3(3(4(0(4(3(3(0(x1)))))))))) (13)
3(2(4(1(2(2(x1)))))) 4(4(4(5(4(2(0(3(4(2(x1)))))))))) (14)
3(2(5(2(2(0(x1)))))) 3(4(4(0(2(1(3(5(3(0(x1)))))))))) (15)
3(4(1(2(2(3(x1)))))) 3(5(4(1(4(4(4(4(4(5(x1)))))))))) (16)
3(5(2(0(3(5(x1)))))) 3(3(4(3(4(3(4(4(2(5(x1)))))))))) (17)
4(0(0(0(1(4(x1)))))) 2(3(0(3(3(4(1(3(1(4(x1)))))))))) (18)
4(0(1(3(2(4(x1)))))) 4(1(4(1(3(3(1(4(1(4(x1)))))))))) (19)
4(3(0(5(4(4(x1)))))) 4(3(4(5(3(1(4(4(2(4(x1)))))))))) (20)
5(1(2(3(5(0(x1)))))) 2(2(4(4(2(4(4(3(3(0(x1)))))))))) (21)
5(3(0(1(3(3(x1)))))) 2(4(4(1(4(4(3(1(3(3(x1)))))))))) (22)
5(4(5(5(5(3(x1)))))) 3(5(4(4(2(4(3(4(0(3(x1)))))))))) (23)
5(5(0(1(5(3(x1)))))) 5(4(4(4(1(3(0(5(4(3(x1)))))))))) (24)
5(5(5(0(2(2(x1)))))) 2(4(1(4(3(4(3(4(4(3(x1)))))))))) (25)
5(5(5(5(2(3(x1)))))) 3(3(0(4(1(4(2(4(4(3(x1)))))))))) (26)
0(2(4(5(2(2(3(x1))))))) 0(5(1(4(4(1(3(1(5(3(x1)))))))))) (27)
0(3(0(4(1(5(3(x1))))))) 0(0(3(3(4(4(3(0(0(5(x1)))))))))) (28)
0(4(3(4(5(2(2(x1))))))) 0(0(4(5(3(4(2(3(3(2(x1)))))))))) (29)
1(3(2(0(2(2(3(x1))))))) 1(3(4(3(5(1(1(1(2(3(x1)))))))))) (30)
1(4(5(5(2(2(0(x1))))))) 3(2(1(3(4(4(5(0(3(0(x1)))))))))) (31)
1(5(0(2(2(2(4(x1))))))) 0(5(1(3(5(4(3(3(1(4(x1)))))))))) (32)
1(5(4(0(2(1(3(x1))))))) 0(1(1(5(3(3(4(4(0(3(x1)))))))))) (33)
2(0(1(5(2(0(5(x1))))))) 3(4(0(0(3(1(3(0(2(5(x1)))))))))) (34)
2(2(0(0(2(2(4(x1))))))) 2(1(1(4(4(5(4(4(4(4(x1)))))))))) (35)
2(3(0(5(0(1(3(x1))))))) 2(3(1(0(5(1(0(3(1(3(x1)))))))))) (36)
2(4(0(2(2(5(0(x1))))))) 1(3(0(4(5(4(4(0(3(0(x1)))))))))) (37)
2(4(5(0(2(5(0(x1))))))) 4(4(3(1(3(4(0(5(1(1(x1)))))))))) (38)
2(5(2(2(5(2(4(x1))))))) 4(3(4(1(3(0(4(0(4(4(x1)))))))))) (39)
3(2(0(2(2(2(2(x1))))))) 4(5(3(1(3(2(3(5(0(5(x1)))))))))) (40)
3(2(5(5(2(4(5(x1))))))) 3(0(0(0(3(2(4(3(4(5(x1)))))))))) (41)
3(2(5(5(3(2(3(x1))))))) 3(1(0(5(0(3(2(4(3(3(x1)))))))))) (42)
3(3(5(0(2(2(2(x1))))))) 4(4(3(0(4(3(3(5(3(5(x1)))))))))) (43)
3(4(5(2(1(1(2(x1))))))) 3(4(5(3(5(4(4(2(0(5(x1)))))))))) (44)
4(0(3(3(1(5(4(x1))))))) 1(3(1(0(0(0(3(4(4(4(x1)))))))))) (45)
4(1(2(4(1(2(2(x1))))))) 3(4(5(3(1(1(4(4(0(5(x1)))))))))) (46)
5(2(5(2(3(3(2(x1))))))) 4(3(1(0(3(1(3(2(5(3(x1)))))))))) (47)
5(3(2(2(3(0(2(x1))))))) 4(5(4(3(3(1(0(5(0(2(x1)))))))))) (48)
5(3(2(5(2(5(0(x1))))))) 5(4(2(2(4(4(3(0(3(1(x1)))))))))) (49)
5(5(2(2(2(2(0(x1))))))) 0(5(3(5(1(3(1(0(3(0(x1)))))))))) (50)
5(5(5(2(1(1(0(x1))))))) 4(2(2(4(5(4(2(3(3(1(x1)))))))))) (51)
5(5(5(2(2(0(0(x1))))))) 1(4(0(3(3(4(2(3(3(1(x1)))))))))) (52)

Property / Task

Prove or disprove termination.

Answer / Result

Yes.

Proof (by AProVE @ termCOMP 2023)

1 String Reversal

Since only unary symbols occur, one can reverse all terms and obtains the TRS
2(1(0(0(x1)))) 3(0(0(3(5(4(4(5(4(3(x1)))))))))) (53)
2(2(2(3(x1)))) 3(5(3(1(4(4(4(2(3(0(x1)))))))))) (54)
2(5(4(2(0(x1))))) 3(1(3(0(0(0(0(4(3(0(x1)))))))))) (55)
1(5(0(4(1(x1))))) 0(3(3(3(4(5(4(4(4(5(x1)))))))))) (56)
0(2(1(4(5(x1))))) 0(3(1(4(0(3(4(4(0(0(x1)))))))))) (57)
5(3(5(2(3(0(x1)))))) 5(3(3(4(3(4(4(3(1(3(x1)))))))))) (58)
0(5(1(1(5(0(x1)))))) 0(3(1(3(3(1(3(0(0(0(x1)))))))))) (59)
3(2(2(5(2(1(x1)))))) 3(3(4(3(4(5(4(2(4(0(x1)))))))))) (60)
3(2(1(0(0(2(x1)))))) 3(4(4(4(3(1(3(3(3(1(x1)))))))))) (61)
1(5(3(1(0(2(x1)))))) 1(0(3(5(4(5(4(3(0(0(x1)))))))))) (62)
1(0(0(5(5(2(x1)))))) 0(3(5(1(3(4(3(0(1(4(x1)))))))))) (63)
1(3(0(4(0(3(x1)))))) 1(3(0(1(0(3(5(4(4(4(x1)))))))))) (64)
0(5(5(1(2(3(x1)))))) 0(3(3(4(0(4(3(3(1(4(x1)))))))))) (65)
2(2(1(4(2(3(x1)))))) 2(4(3(0(2(4(5(4(4(4(x1)))))))))) (66)
0(2(2(5(2(3(x1)))))) 0(3(5(3(1(2(0(4(4(3(x1)))))))))) (67)
3(2(2(1(4(3(x1)))))) 5(4(4(4(4(4(1(4(5(3(x1)))))))))) (68)
5(3(0(2(5(3(x1)))))) 5(2(4(4(3(4(3(4(3(3(x1)))))))))) (69)
4(1(0(0(0(4(x1)))))) 4(1(3(1(4(3(3(0(3(2(x1)))))))))) (70)
4(2(3(1(0(4(x1)))))) 4(1(4(1(3(3(1(4(1(4(x1)))))))))) (71)
4(4(5(0(3(4(x1)))))) 4(2(4(4(1(3(5(4(3(4(x1)))))))))) (72)
0(5(3(2(1(5(x1)))))) 0(3(3(4(4(2(4(4(2(2(x1)))))))))) (73)
3(3(1(0(3(5(x1)))))) 3(3(1(3(4(4(1(4(4(2(x1)))))))))) (74)
3(5(5(5(4(5(x1)))))) 3(0(4(3(4(2(4(4(5(3(x1)))))))))) (75)
3(5(1(0(5(5(x1)))))) 3(4(5(0(3(1(4(4(4(5(x1)))))))))) (76)
2(2(0(5(5(5(x1)))))) 3(4(4(3(4(3(4(1(4(2(x1)))))))))) (77)
3(2(5(5(5(5(x1)))))) 3(4(4(2(4(1(4(0(3(3(x1)))))))))) (78)
3(2(2(5(4(2(0(x1))))))) 3(5(1(3(1(4(4(1(5(0(x1)))))))))) (79)
3(5(1(4(0(3(0(x1))))))) 5(0(0(3(4(4(3(3(0(0(x1)))))))))) (80)
2(2(5(4(3(4(0(x1))))))) 2(3(3(2(4(3(5(4(0(0(x1)))))))))) (81)
3(2(2(0(2(3(1(x1))))))) 3(2(1(1(1(5(3(4(3(1(x1)))))))))) (82)
0(2(2(5(5(4(1(x1))))))) 0(3(0(5(4(4(3(1(2(3(x1)))))))))) (83)
4(2(2(2(0(5(1(x1))))))) 4(1(3(3(4(5(3(1(5(0(x1)))))))))) (84)
3(1(2(0(4(5(1(x1))))))) 3(0(4(4(3(3(5(1(1(0(x1)))))))))) (85)
5(0(2(5(1(0(2(x1))))))) 5(2(0(3(1(3(0(0(4(3(x1)))))))))) (86)
4(2(2(0(0(2(2(x1))))))) 4(4(4(4(5(4(4(1(1(2(x1)))))))))) (87)
3(1(0(5(0(3(2(x1))))))) 3(1(3(0(1(5(0(1(3(2(x1)))))))))) (88)
0(5(2(2(0(4(2(x1))))))) 0(3(0(4(4(5(4(0(3(1(x1)))))))))) (89)
0(5(2(0(5(4(2(x1))))))) 1(1(5(0(4(3(1(3(4(4(x1)))))))))) (90)
4(2(5(2(2(5(2(x1))))))) 4(4(0(4(0(3(1(4(3(4(x1)))))))))) (91)
2(2(2(2(0(2(3(x1))))))) 5(0(5(3(2(3(1(3(5(4(x1)))))))))) (92)
5(4(2(5(5(2(3(x1))))))) 5(4(3(4(2(3(0(0(0(3(x1)))))))))) (93)
3(2(3(5(5(2(3(x1))))))) 3(3(4(2(3(0(5(0(1(3(x1)))))))))) (94)
2(2(2(0(5(3(3(x1))))))) 5(3(5(3(3(4(0(3(4(4(x1)))))))))) (95)
2(1(1(2(5(4(3(x1))))))) 5(0(2(4(4(5(3(5(4(3(x1)))))))))) (96)
4(5(1(3(3(0(4(x1))))))) 4(4(4(3(0(0(0(1(3(1(x1)))))))))) (97)
2(2(1(4(2(1(4(x1))))))) 5(0(4(4(1(1(3(5(4(3(x1)))))))))) (98)
2(3(3(2(5(2(5(x1))))))) 3(5(2(3(1(3(0(1(3(4(x1)))))))))) (99)
2(0(3(2(2(3(5(x1))))))) 2(0(5(0(1(3(3(4(5(4(x1)))))))))) (100)
0(5(2(5(2(3(5(x1))))))) 1(3(0(3(4(4(2(2(4(5(x1)))))))))) (101)
0(2(2(2(2(5(5(x1))))))) 0(3(0(1(3(1(5(3(5(0(x1)))))))))) (102)
0(1(1(2(5(5(5(x1))))))) 1(3(3(2(4(5(4(2(2(4(x1)))))))))) (103)
0(0(2(2(5(5(5(x1))))))) 1(3(3(2(4(3(3(0(4(1(x1)))))))))) (104)

1.1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.

There are 497 ruless (increase limit for explicit display).

1.1.1 Semantic Labeling Processor

The following interpretations form a model of the rules.

As carrier we take the set {0,1}. Symbols are labeled by the interpretation of their arguments using the interpretations (modulo 2):

[5(x1)] = 0
[3#(x1)] = 0
[2#(x1)] = 0
[0#(x1)] = 0
[5#(x1)] = 0
[4#(x1)] = 0
[1#(x1)] = 0
[0(x1)] = 0
[1(x1)] = 0
[2(x1)] = 0
[3(x1)] = 0
[4(x1)] = 1

We obtain the set of labeled pairs

There are 994 ruless (increase limit for explicit display).

and the set of labeled rules:

There are 104 ruless (increase limit for explicit display).

1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.